
Journal of Convex Analysis 26 (2019), No. 2, 515526 Copyright Heldermann Verlag 2019 Smoothness in some Banach Spaces of Operators and Vector Valued Functions Enrique Jordá Dep. de Matemática Aplicada, Universidad Politécnica de Valencia, Plaza Ferrándiz y Carbonell 2, 03801 Alcoy, Spain ejorda@mat.upv.es Ana María Zarco Dep. de Matemática Aplicada, Universidad Politécnica de Valencia, Plaza Ferrándiz y Carbonell 2, 03801 Alcoy, Spain anzargar@upv.es [Abstractpdf] A well known criterion of \v{S}mulyan states that the norm $\\cdot\$ of a real Banach space $X$ is G\^{a}teaux differentiable at $x\in X$ if and only if there is $x^*\in S_{X^*}$ which is $w^*$exposed by $x$ in $B_{X^*}$ and that the norm is Fr\'echet differentiable at $x$ if and only if there is $x^*\in S_{X^*}$ which is $w^*$strongly exposed in $B_{X^*}$ by $x$. We show that in this criterion $B_{X^*}$ can be replaced by a convenient smaller set, and we apply this extended criterion to characterize the points of G\^{a}teaux and Fr\'echet differentiability of the norm in epsilon products of Banach spaces, extending previous work of Heinrich. As a consequence we get some results of smoothness of the norm in some Banach spaces of continuous and harmonic vector valued functions. Keywords: Banach spaces, Frechet and Gateaux differentiability, epsilon products. MSC: 46B20, 46B50 [ Fulltextpdf (134 KB)] for subscribers only. 